The real problem here is not Airbnb, which is a symptom; it's zoning, which forbids market-clearing prices and a sufficient number of units to satisfy demand for living in and visiting the city.
For more on this, see e.g. Matt Yglesias's The Rent is Too Damn High: https://www.amazon.com/Rent-Too-Damn-High-Matters-ebook/dp/B... .
When we have sane zoning, we'll have sane prices. Today there was an NYT opinion piece that comes at the issue from a racial angle: https://www.nytimes.com/2017/08/03/opinion/sunday/zoning-law... but I don't think we even need the racial angle; we can come at it from a position of fundamental justice and human flourishing.